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HP Forum Archive 17

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Re: Significant digits -- well, yes and no...
Message #1 Posted by Massimo Gnerucci (Italy) on 21 Feb 2007, 2:49 a.m.

Quote:
HP-42S: sin (3.14159265358 rad) = 9.79323846264 x 10-12 -- correct result to 12 significant digits

HP-41: sin (3.141592653 rad) = 5.9 x 10-10 -- correct result to 2 significant digits

Sorry Karl, shouldn't that be:
HP-41: sin (3.141592654 rad) = -4.1 x 10-10 vs -4.10206761537 x 10-10 ?

Greetings,
Massimo

      
Missing pi digits calculation
Message #2 Posted by Karl Schneider on 21 Feb 2007, 3:47 a.m.,
in response to message #1 by Massimo Gnerucci (Italy)

Hi, Massimo --

Quote:
Sorry Karl, shouldn't that be: HP-41: sin (3.141592654 rad) = -4.1 x 10-10 vs -4.10206761537 x 10-10 ?

That also is a correct calculation, but my point was to reveal the ensuing digits of pi by calculating a truncated (not rounded) value of pi in radians mode. I've gone through the exercise several times in the Forum, but didn't save a bookmark to those posts:

sin(pi - x) = sin(pi)*cos(x) - cos(pi)*sin(x)

            =    0 * cos(x)  -  (-1)*sin(x)

            =                     sin(x)

x represents the truncated digits. For very small x, sin(x) ~= x, so the result produces a limited string of those digits.

The excellent mathematical routines developed for the Saturn microprocessor (debuting with the HP-71B) were ported to the Pioneer-series calculators. No other calculator I own matches the quality of the Saturn mathematics, although the TI-89 might. It also seems likely that Valentin's vintage Sharp pocket computers could meet or exceed the accuracy.

-- KS

            
SHARP accuracy
Message #3 Posted by Valentin Albillo on 21 Feb 2007, 5:15 p.m.,
in response to message #2 by Karl Schneider

Hi, Karl:

Karl posted:

    "It also seems likely that Valentin's vintage Sharp pocket computers could meet or exceed the accuracy."

      Likely. These are your results as computed by my SHARP PC-1475, rounded to 12 digits:

             sin (3.14159265358 rad) = 9.79323846265E-12
      
       sin (3.141592653 rad)   = 5.89793238463E-10
      
       cos (89.9999999 deg)    = 1.74532925199E-09

      Matter of fact, this last result actually comes out in full precision as:

             cos (89.9999999 deg) = 1.7453292519943295760D-09
      which has all its 20 significant digits absolutely correct.

Best regards from V.

                  
Re: SHARP accuracy
Message #4 Posted by Karl Schneider on 23 Feb 2007, 1:39 a.m.,
in response to message #3 by Valentin Albillo

Hi, Valntin --

Quote:
These are your results as computed by my SHARP PC-1475, rounded to 12 digits:

sin (3.14159265358 rad) = 9.79323846265E-12

But, V! The last mantissa digit should be a "4", whether it's the actual digit or rounded. Who was responsible for the rounding? :-)

Seriously though, 20 significant digits was quite impressive for the era. I safely considered the 12+ digit performance "likely" from your photos I had previously viewed, at least one of which depicted a Sharp "PC" with many digits showing in its display.

Best regards,

-- KS

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